RUI: Pure and Applied Knot Theory: Skeins, Hyperbolic Volumes, and Biopolymers

结理论是对环的纠缠直至连续变形的数学研究。一个人可以通过将缠绕的绳子的两端连接起来来打结，如果一个人可以不断地将一根绳变形为另一根绳，例如通过弯曲、拉伸和将绳子穿过其他绳子，但不以任何方式切断或折断绳子，那么两个结就是等效的。本项目考虑数学结理论的理论问题和应用。一组问题研究与物理学量子场论有关的结的不变量族。更具体地说，该项目旨在了解结的量子不变量如何检测结的几何特性以及与之相关的三维空间。本研究的数学技术在数学物理和理论拓扑量子计算中具有潜在的应用前景。另一组问题涉及到研究结蛋白和其他生物聚合物的应用，其中一些已知与各种疾病有关。该项目使用结理论技术来开发一个模型，该模型可用于量化并将局部拓扑复杂性与生物物理过程联系起来。该模型还可以用于设计具有特殊生物物理性质的合成生物聚合物。该项目包括一些适合与本科生合作的研究问题，以及旨在提高人们对数学兴趣的推广和传播活动。PI过去曾成功地让本科生参与类似的研究，并将继续建议和鼓励学生继续从事数学和相关领域的职业。这项研究分为三个部分，两个部分寻求将量子拓扑与双曲几何联系起来，一个部分将结理论应用于分子生物学。其中一个项目涉及基于量子拓扑中的Kauffman托架串代数理论的体积猜想版本，以及它与双曲几何中曲面的Teichmuller空间的关系。第二个课题研究了与带穿孔曲面的修饰Teichmuller空间有关的Kauffman托架代数的一种推广的代数和几何性质。第三个项目是与一位生物物理学家合作，研究生物聚合物中被分子力紧紧束缚在一起的局部缠结。提出的结理论模型将给出这种局部纠缠的描述，允许人们在实验中量化和测量生物聚合物局部拓扑复杂性的变化。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Knot theory is the mathematical study of entanglement of loops up to continuous deformation. One can create a knot by taking an entangled string and connecting the endpoints, and two knots are equivalent if one can continuously deform one to the other, for example by bending, stretching, and passing strands inside and through others, but without cutting or breaking the string in any way. This project considers both theoretical problems and applications of mathematical knot theory. One group of problems studies a family of invariants of knots related to quantum field theory from physics. More specifically, the project seeks to understand how the quantum invariant of a knot detects geometric properties of the knot and the 3-dimensional spaces that can be associated with it. The mathematical techniques from this research has potential applications to mathematical physics and theoretical topological quantum computing. Another group of problems concerns applications to the study of knotted proteins and other biopolymers, some of which are known to be associated to various diseases. The project uses knot theory techniques to develop a model that can be used to quantify and to relate local topological complexity with biophysical processes. The model can also be used to potentially design synthetic biopolymers with special biophysical properties. The project includes a number of research problems suitable for collaboration with undergraduate students, as well as outreach and dissemination activities that seek to increase interest in mathematics more generally. The PI has successfully involved undergraduate students in similar research in the past and will continue to advise and encourage students to continue careers in mathematics and related areas. The research is split into three parts, two seek to connect quantum topology with hyperbolic geometry and one applies knot theory to molecular biology. One project concerns a version of the Volume Conjecture based the theory of the Kauffman bracket skein algebra from quantum topology and its relationship to the Teichmuller space of a surface from hyperbolic geometry. A second project studies algebraic and geometric properties of a generalization of the Kauffman bracket algebra which is related to the decorated Teichmuller space of a surface with punctures. A third project involves a collaboration with a biophysicist to study local entanglements that are held tightly in place by molecular forces in biopolymers. The proposed knot-theoretic model would give a description of such local entanglements, allowing one to quantify and measure changes in the local topological complexity of biopolymers in experiments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.